The effective management of geo-energy systems heavily relies on robust modeling frameworks that integrate diverse simulation capabilities, including flow and transport, phase equilibrium, geochemistry and geomechanics. While a multiphysics simulation engine within a unified framework has its advantages, integrating specialized modeling packages often enhances viability. Efficient and seamless communication between these engines be- comes crucial for improving the performance and scalability of the integration. Advanced parametrization tech- niques can facilitate this integration by efficiently approximating and interpolating coupling data, ensuring both speed and accuracy. In this study, we compare the efficiency of different interpolation techniques used for the parametrization of complex many-component fluid systems in compositional simulation. We employ an Operator- Based Linearization (OBL) framework that leverages the general formulation of corresponding conservation laws. OBL effectively learns the operators required for assembly of the laws while interpolation delivers fast evalua- tion of operators and their derivatives for all physical states in a simulation domain. Multilinear interpolation is a simple and robust approach, yet it has poor scaling properties with respect to the dimension of the physical state. To alleviate interpolation costs in multiple dimensions, we study the performance and accuracy of other interpolation techniques, including linear interpolation with standard and Delaunay triangulation. Overall, this approach provides great flexibility, saves development costs and simplifies the incorporation of thermodynamics and geochemistry engines for precise modeling of phase equilibrium, reactive transport, dissolution-precipitation and kinetics of chemical reactions. This research extends the scalability of the OBL framework and addresses the challenges of high dimensionality in compositional modeling. Consequently, this approach holds significant potential for integrating various complex multiphysics problems, enabling the creation of more comprehensive digital twins for geo-energy systems management.

The SPE11 comparative solution project presents a benchmark for geological carbon storage in an aquifer, as the development of sufficiently accurate CO₂ sequestration models is critical for predicting the distribution of CO₂ during and after injection. In this paper we present a convergence analysis of the SPE11 benchmark simulation using the Delft Advanced Research Terra Simulator (open-DARTS). Open-DARTS, an open-source simulation framework designed for forward and inverse modeling, as well as uncertainty quantification, employs a unified thermal-compositional formulation and operator-based linearization. In our convergence analysis the SPE11b (2D - reservoir conditions) starts to converge at a grid resolution of 1340 × 240, after which added resolution provides diminishing returns. In addition the three-dimensional SPE11c benchmark is simulated with 8M grid blocks. However, 2D results from SPE11b suggest that a greater resolution is required for a truly converged solution. Furthermore, we extend the SPE11b benchmark to include H₂S as a trace impurity in the injection stream.

The open Delft Advanced Research Terra Simulator (open-DARTS) framework is an open-source reservoir simulation software. The open-DARTS focused on energy transition applications, such as geothermal energy production and carbon sequestration. It enables the modeling of compositional thermal flow, coupled with a geomechanical solver based on the Finite Volume discretization and adjoints method for inverse modeling. The open-DARTS supports different grid types (structured, corner-point geometry, and unstructured), discrete fracture networks, contact mechanics, and various thermal-chemical interactions. The recently proposed generic nonlinear formulation supports the most general nonlinear PDEs designed for various energy transition applications. The open-DARTS has been implemented in C++ and Python to optimize hardware utilization while ensuring flexibility. The most computationally expensive part is written in C++ and compiled into libraries, which are subsequently exposed to Python using pybind11. This allows the extension and overriding of C++ functions by user-defined Python code. For example, using only a Python interface, one can adjust a timestep strategy, nonlinear solver, or properties output. Besides, the Python interface of open-DARTS provides straightforward coupling with other Python-based numerical modeling packages, including the meshing, file storage, caching, and visualization modules. The open-DARTS core uses the advantages of C++ language, such as efficient low-level memory management, object-oriented programming, compile-time polymorphism, and parallelization with OpenMP. One of the advantages of open-DARTS is the Operator-Based Linearization (OBL) technique, which can resolve challenges associated with complex physics and reduce the computation time, especially for ensemble-based simulations. We would also like to share our experience on the project, repository, and the development workflow configuration using gitlab.com, including the build system (cmake), handling merge requests, automated testing in CI/CD pipelines, documentation management (gitlab.io), wiki utilization, and release publishing. Additionally, Python’s integration into open-DARTS offers the advantage of straightforward installation via PyPI and simplifies defining requirements for users who prefer to avoid compiling code from source files.

The role of Thermal-Hydro-Mechanical-Compositional analysis in the development of geo-energy resources has been amplified in recent years. As an example, challenges such as wellbore stability, land subsidence and induced seismicity highlight the necessity for comprehensive geomechanical evaluations which are then coupled with thermo-hydrodynamical processes within the reservoir. Numerical simulations of the coupled thermo-poromechanical processes provide a general-purpose tool capable of performing these evaluations at both continuum laboratory and field scales. However, efficient integration of the coupled system of fluid mass, energy and momentum conservation equations poses multiple numerical and implementation difficulties, such as combining different numerical methods on staggered grids and associated limitations on admissible grids. This paper introduces a new fully-implicit scheme of the Finite Volume Method (FVM) for modeling thermal compositional flow in thermo-poroelastic rocks. The scheme uses the gradient-based variant of coupled multi-point approximations of fluid mass, momentum, heat convection and conduction fluxes, which are derived from their respective local balances. The novelty of the scheme is that it incorporates temperature into the approximation of these fluxes. Consequently, the approximation of displacement gradients depends on temperatures, while the approximation of temperature itself is derived from the balance of heat conduction fluxes. At the same time, we utilize a single-point upstream weighting for the temperature-dependent terms in heat convection fluxes. The resulting scheme respects the local balance of fluxes in the presence of temperature gradients. Besides, it also supports star-shaped and various boundary conditions. Overall, the scheme represents a unified FVM-based approach for the integration of all conservation laws relevant to geo-energy applications on a cell-centered collocated grid. Furthermore, the implemented two-stage block-partitioned preconditioning strategy enables the efficient solution of obtained linear systems. The proposed modeling framework has been implemented in an open-source Delft Advanced Research Terra Simulator (DARTS). Moreover, the flexibility regarding compositional fluid properties is reinforced by the Operator-Based Linearization (OBL) technique incorporated into DARTS. The proposed modeling framework has undergone rigorous validation in convergence study, and comparisons against established analytical and numerical solutions. The framework covers advanced physical phenomena including thermal expansion and contraction, porosity dependent on pressure, temperature and strain, and multiphase flow with phase changes and chemical alterations. The framework capabilities and the performance of the preconditioning strategy have been assessed in the mechanical extension of the 10th SPE Comparative study (SPE10) model.

In this work, we describe our decisions made to perform the FluidFlower simulation study and discuss various aspects of the benchmark that are different from our usual subsurface simulation practice. We will discuss the impact of various modeling choices on the outcomes of the simulation models, such as gridding, discretization, and solver strategies, and the lessons learned, taking into account the different conditions of the FluidFlower study compared to conditions commonly dealt with in subsurface simulation. We will start with a brief description of the DARTS framework utilized for compositional simulation, the thermodynamic and physical modeling related to the atmospheric CO2 -brine system, and the modeling workflow used in our benchmark submission. Additionally, we describe a custom nonlinear solver developed for the atmospheric benchmark conditions to improve convergence including the linear solver strategy since our default two-stage preconditioner does not perform effectively. To make meaningful comparisons between each of the modeling choices, we define a baseline model which is a simplified version of our setup in the main FluidFlower benchmark. The baseline model is then used to study the effect of Cartesian and unstructured meshes and a two-point flux approximation compared with a multi-point flux approximation for capturing the physics at play. We conclude our work with lessons learned and future recommendations.

Elliptic differential operators describe a wide range of processes in mechanics relevant to geo-energy applications. Extensively used in reservoir modeling, the Finite Volume Method with TPFA can be consistently applied to discretize only a specific type of application under severe assumptions. In this paper, we introduce a positivity preserving Nonlinear Two Point Stress Approximation (NTPSA) based on the recently developed collocated Finite Volume scheme for linear elastic mechanics. The gradient reconstruction is different from the one used in Nonlinear TPFA, but a similar form of weighting scheme is employed to reconstruct the traction vector at each interface. The convergence of the scheme is tested with a homogeneous anisotropic stiffness tensor. The motivation behind the implementation of a new discretization framework in mechanics is to develop a uniform discretization technique preserving monotonicity for generic poromechanics applications.

Quantification of the poromechanical response of subsurface formations due to human-induced pore pressure fluctuations is critical for the performance and stability assessment of many geo-energy systems. In particular, natural faults in the subsurface introduce the hazard of induced seismicity. Numerical modeling of fault reactivation is challenging, while the specific details of induced stresses and fault slip in reservoirs with displaced (i.e. non-zero offset) faults may cause additional challenges depending on the type of numerical formulation employed. To facilitate the systematic development and testing of numerical tools for the simulation of induced seismicity in faulted reservoirs we developed a set of semi-analytical test problems of increasing complexity, based on inclusion theory and Cauchy singular integral equations. With these we investigate the accuracy of two recently developed Finite Volume (FV) schemes with collocated and staggered arrangements of unknowns. One of them employs a conformal discrete fault model (DFM) which can guarantee sufficient accuracy at the cost of adaptive mesh refinement but may suffer from modelling and computational challenges when addressing large-scale realistic geological configurations. The second one employs an embedded (or non-conformal) discrete fault model (EDFM) which avoids the need for excessive mesh refinement, but of which the accuracy and the range of applicability are still to be investigated. We found that both numerical schemes accurately represent the pre-slip Coulomb stresses, but show different degrees of accuracy in representing the resulting depletion-induced fault slip. The semi-analytical benchmark data are available via DOI 10.4121/22240309.

We present a scalable collocated Finite Volume Method (FVM) to simulate induced seismicity as a result of pore pressure changes. A discrete system is obtained based on a fully-implicit fully-coupled description of flow, elastic deformation, and contact mechanics at fault surfaces on a flexible unstructured mesh. The cell-centered collocated scheme leads to a convenient integration of the different physical equations, as the unknowns share the same discrete locations on the mesh. Additionally, a generic multi-point flux approximation is formulated to treat heterogeneity, anisotropy, and cross-derivative terms for both flow and mechanics equations. The resulting system, though flexible and accurate, can lead to excessive computational costs for field-relevant applications. To resolve this limitation, a scalable processing algorithm is developed and presented. Several proof-of-concept numerical tests, including benchmark studies with analytical solutions, are investigated. It is found that the presented method is indeed accurate and efficient; and provides a promising framework for accurate and efficient simulation of induced seismicity in various geoscientific applications.

We develop a collocated Finite Volume Method (FVM) to study induced seismicity as a result of pore pressure fluctuations. A discrete system is obtained based on a fully-implicit coupled description of flow, elastic deformation, and contact mechanics at fault surfaces on a fully unstructured mesh. The cell-centered collocated scheme leads to convenient integration of the different physical equations, as the unknowns share the same discrete locations on the mesh. Additionally, a multi-point flux approximation is formulated in a general procedure to treat heterogeneity, anisotropy, and cross-derivative terms for both flow and mechanics equations. The resulting system, though flexible and accurate, can lead to excessive computational costs for field-relevant applications. To resolve this limitation, a scalable parallel solution algorithm is developed and presented. Several proof-of-concept numerical tests, including benchmark studies with analytical solutions, are investigated. It is found that the presented method is indeed accurate, stable and efficient; and as such promising for accurate and efficient simulation of induced seismicity.